Ok, here goes. I have a real problem with understanding a certain thing I was thinking about today in school. Now, keep in mind I'm not a math major, I'm just some moron in high school.
Anyway, here's the thing causing confusion:
Let's say that we have a function and we graph it. This function, f(x), is defined as x - 1. Graphing it results in what you would expect: A line, slope of 1, crossing the y axis at -1. Simple enough so far.
Now, we all probably know that when you multiply something by 1, you get the original something. Also, something divided by itself it 1.
Using these two properties or theorems or whatever who's names I don't know 1, we shall proceed as follows:
f(x) = (x - 1)
= 1(x - 1)
(x + 1)
= ------- (x - 1)
(x + 1)
(x2 - 1)
(x + 1)
(Pardon my attempt at formatting if it looks bad on your computer)
So we've arrived at a new way of stating our original f(x). Indeed, should you graph this new one, it will look exactly like the first one. Exactly, with one small problem... f(-1) now works out to zero divided by zero, where before it was -2. They haven't decided what zero divided by zero is yet, but I'm fairly certain that "-2" isn't one of the major proposals...2
In case it's hard to tell (which it is), my confusion is not about 0/0 per se. It's that some of our more basic rules of math (seemingly, at least) allow us to make such a transformation.
Yes, it all boils down to questions involving 0/0. And I am pretty sure that this confuses everybody.
The real issue is that all my life (which, as I am 17, is mostly in schools) I've known math to be a true/false test in a world of essays. 2 + 2 = 4, no subjectivity involved. And then today in Calculus, as we were learning about limits, I come to comprehend that even math is not black and white. Math, in my mind, went to shades of gray. Then to tie-dye. What in the world is this, that would make the huge gaping maw of weirdness appear in the middle of a structured, constant, and ordered thing such as math?
But, as I've said, I am just a teenager who's taken some math classes, and should anyone see any glaring errors in my way of thinking this through, please, please, please point them out. I'd really like to know what is going on.
1: Man, I knew I should have paid attention in math class, instead of playing games on my TI-86...
2: I just noticed that this entire thing is touched upon, perhaps more indirectly than not, by a couple of folks in the zero divided by zero node. Go forth and be slightly more informed. It's still all up in the air, though, really...
Looking through my message inbox, I notice that several people have pointed out to me that the "solution" to this little problem lies in the fact that my fraction up there is equal to one only when x is not equal to -1. Thus, it should not be substituted for 1.
Doesn't really make the problem go away, though. If it were in a large fractions with many variables and factors, most people would just cross them both out. Why? 'Cause it's equal to one...
...until you start to look closer.